Stochastic Averaging Technique Research Articles

Random Vibration Analysis of Double Deck Rocking Self-Centering Pier Under Seismic Excitation

This paper investigates nonlinear random vibration of double deck rocking self-centering pier structure subjected to seismic excitation. First of all, a novel type of double deck rocking self-centering pier that offers better resilience is designed. The stochastic dynamical model of the system is then established, in which the seismic excitation is viewed as Kanai–Tajimi filtered white noise and the self-centering restoring force is characterized by classical flag-shaped hysteretic model. Subsequently, the self-centering restoring force is decomposed into equivalent quasi-linear elastic and damping forces by adopting the harmonic balance (HB) scheme. Following this, the stochastic averaging (SA) technique is implemented to derive the averaged Itô equation. The steady-state response probability density with respect to the amplitudes is solved from the averaged Fokker–Planck–Kolmogorov (FPK) equation. Finally, the effects of system parameters on the stochastic response of the double deck rocking self-centering pier structure are performed, and validated by the Monte Carlo simulation (MCS). In addition, with the help of the Laplace transform, the transition function as well as conditional power spectral density are achieved, which are combined with the steady-state probability density to obtain the power spectral density response. This research will contribute to the optimal seismic design of double-deck rocking self-centering piers.

An improved coupled tri-stable energy harvesting system with spring stops for passive control

Motivated by enhancing the DC power output of a tri-stable energy harvester (TEH), a passive control of a coupled TEH driven by colored noise is constructed by using spring stops. The cubic trilinear spring model is utilized to describe the piecewise nonlinear impact force introduced by the spring stops. Using the stochastic averaging technique based on energy-dependent frequency, the analytical expressions of joint probability density function (PDF) and harvested power are obtained. The effects of the impact distance, impact stiffness, and colored noise on stochastic dynamics and harvesting performance are further discussed. It is found that the spring stops can change the shape of the PDF to induce stochastic bifurcation phenomena. The TEH under passive control tends to oscillate among three potential wells to effectively improve the DC output. By adjusting the appropriate impact distance and stiffness, the harvesting performance can be optimized and improved. Moreover, the numerical simulations and experimental verifications well support the theoretical analysis. Based on the experimental results, the harvested power can be enhanced by choosing the appropriate material of the stop (i.e., the spring is the optimal one among three different materials).

Stochastic dynamics of a nonlinear vibration energy harvester subjected to a combined parametric and external random excitation: The distinct cases of Itô and Stratonovich stochastic integration

The Stratonovich and Itô interpretations of a stochastic integral are mathematically consistent, distinct representations that yield contrasting response characteristics for stochastic dynamical systems subjected to multiplicative noise, such as parametric random excitation. The contrasting dynamics from the two interpretations currently remain unexplored in vibration energy harvester dynamics. We analytically investigate the dynamics of a generic piezoelectric, nonlinear vibration energy harvester simultaneously subjected to stochastic damping and external random excitation, focusing on the contrasting dynamics engendered by the two interpretations. Numerically solving the Stratonovich and Itô stochastic differential equations in the nonequilibrium regime of harvester dynamics, we find that the Stratonovich version yields significantly higher root mean square values of the harvester voltage output, for lower values of the mechanical damping coefficient. Furthermore, we find positive values for the leading Lyapunov exponent indicating instabilities in the harvester response unique to the Stratonovich interpretation. Additionally, comparing the results for the harvester electrical output between the cases of external excitation represented by the white noise and the Ornstein–Uhlenbeck noise processes, we find lower root mean square values of the voltage output in the latter case. Moreover, the results indicate that the averaged voltage output becomes progressively lower with increasing correlation time of the Ornstein–Uhlenbeck process. Studying the equilibrium dynamics by deriving Fokker–Planck equations for the response amplitude of the harvester using the harmonic balance method and stochastic averaging techniques, we solve for the stationary probability densities to find that the harvester is more likely to attain a well-defined equilibrium state under the Itô interpretation. In summary, the results contribute to the theoretical understanding of the fundamental contrast between the Stratonovich and Itô interpretations in vibration energy harvester dynamics and are expected to motivate advances in harvester design for various applications.

Model order reduction and stochastic averaging for the analysis and design of micro-electro-mechanical systems

Electro-mechanical systems are key elements in engineering. They are designed to convert electrical signals and power into mechanical motion and vice-versa. As the number of networked systems grows, the corresponding mathematical models become more and more complex, and novel sophisticated techniques for their analysis and design are required. We present a novel methodology for the analysis and design of electro-mechanical systems subject to random external inputs. The method is based on the joint application of a model order reduction technique, by which the original electro-mechanical variables are projected onto a lower dimensional space, and of a stochastic averaging technique, which allows the determination of the stationary probability distribution of the system mechanical energy. The probability distribution can be exploited to assess the system performance and for system optimization and design. As examples of application, we apply the method to power factor correction for the optimization of a vibration energy harvester, and to analyse a system composed by two coupled electro-mechanical resonators for sensing applications.

Supervision by Denoising.

Learning-based image reconstruction models, such as those based on the U-Net, require a large set of labeled images if good generalization is to be guaranteed. In some imaging domains, however, labeled data with pixel- or voxel-level label accuracy are scarce due to the cost of acquiring them. This problem is exacerbated further in domains like medical imaging, where there is no single ground truth label, resulting in large amounts of repeat variability in the labels. Therefore, training reconstruction networks to generalize better by learning from both labeled and unlabeled examples (called semi-supervised learning) is problem of practical and theoretical interest. However, traditional semi-supervised learning methods for image reconstruction often necessitate handcrafting a differentiable regularizer specific to some given imaging problem, which can be extremely time-consuming. In this work, we propose "supervision by denoising" (SUD), a framework to supervise reconstruction models using their own denoised output as labels. SUD unifies stochastic averaging and spatial denoising techniques under a spatio-temporal denoising framework and alternates denoising and model weight update steps in an optimization framework for semi-supervision. As example applications, we apply SUD to two problems from biomedical imaging-anatomical brain reconstruction (3D) and cortical parcellation (2D)-to demonstrate a significant improvement in reconstruction over supervised-only and ensembling baselines. Our code available at https://github.com/seannz/sud.

Nonlinear random vibration of the slender deep-water pier under seismic excitation

The present study investigates the nonlinear random vibration of the deep-water pier exposed to horizontal seismic excitation. First of all, the stochastic dynamic model of the pier is formulated. During the process, the pier is simplified as a cantilever beam fixed on the rock foundation, the seismic excitation is treated as Gaussian white noise, the hydrodynamic pressure is described with the radiation wave theory, and the equation for the nonlinear kinematic of the pier is deduced by the means of Kane’s method. Then, with the application of the stochastic averaging (SA) technique, the Fokker–Plank–Kolmogorov (FPK) equation governing the transient probability density function (PDF) of the amplitude envelope and the backward Kolmogorov (BK) equation for the conditional reliability function (CRF) are derived, respectively. The closed-form stationary PDF can be yielded directly from the reduced FPK equation, while the CRF and the conditional PDF of first-passage time are given after solving the BK equation numerically. Numerical discussions are performed to illustrate the trend of excitation intensity, mass ratio, immersion ratio, and inner and outer hydrodynamic effect on the stationary response and first-passage failure are examined, respectively. It has been shown that increases in the excitation intensity, mass ratio, and immersion ratio can amplify the response and reduce the reliability of the deep-water pier system. The hydrodynamic effect also leads to an amplification of the system response and a reduction in the reliability of the system. Similarly, the presence of inner water can also exacerbate these effects, and this phenomenon becomes more pronounced with increasing immersed ratios. Additionally, the analytical solution is validated by the result obtained by pertained Monte Carlo simulations (MCS). It is noted that this work will be helpful for the optimal seismic design of deep-water piers.

Discussion on applicability of the complex fractional moment method in Hamiltonian system

In this paper, the complex fractional moment (CFM) method is extended from the Langevin system to the Hamiltonian system for the first time, the applicability and advantages of the CFM method in Hamiltonian systems are investigated. According to the quasi-periodic property, the stochastic averaging technique is applied to the stochastic quasi-nonintegrable Hamiltonian system, and the one-dimensional Hamiltonian stochastic differential equation is obtained. By Mellin transformation, the associated Fokker Planck Kolmogorov (FPK) equation governing the Hamilton function is transformed into a set of Ordinary Differential Equations (ODEs) of CFM. The semi-analytical solutions of the FPK equation are obtained by solving the ODEs. Furthermore, by a polynomial approximation method for the Hamiltonian case proposed in this paper, the numerical results show the influence of parameters and initial values on the evolution of the transient probability density functions (PDF) of the system. The accuracy of CFM method in the Hamiltonian case is verified by error analysis.

Stochastic analysis of a hybrid energy harvester with viscoelastic material

The hybrid energy conversion mechanism can improve the performance of the energy harvester compared to the single energy conversion mechanism. Meanwhile, the use of viscoelastic materials can also improve the energy harvesting efficiency, but there is not much research on hybrid energy harvesting systems with viscoelastic material. Different from existing research, this manuscript investigates the probabilistic response and stochastic bifurcation of a hybrid piezoelectric and electromagnetic energy harvester(HPEEH) with viscoelastic term under random excitation. Firstly, the viscoelastic element is approximately equivalent to the linear combination of damping term and stiffness term through variable transformation. Secondly, the accuracy of the proposed method is verified by comparing the numerical solution with the analytical solution using the stochastic averaging technique to solve the equivalent system. Furthermore, the bifurcation conditions for amplitude steady-state probability density are given according to the singularity theory and the definition of the extreme point. It is found that changes in viscoelastic parameters, damping coefficients and electromagnetic coupling coefficients can induce stochastic P-bifurcation. Finally, the effects of system parameters on mean square current, mean square voltage and mean output power are explored.

Application of Complex Fractional Moment in nonlinear system with Gaussian colored noise

This paper studies the transient response of the nonlinear system with Gaussian colored noise. The one-dimensional Itô stochastic differential equation with respect to amplitude response is obtained by the stochastic averaging technique, and the Mellin transformation is applied to the related Fokker–Planck–Kolmogorov (FPK) equation to obtain a set of Complex Fractional Moment (CFM) differential equations. The semi-analytical solution of the FPK equation can be reconstructed by solving the CFM equations. Numerical method is adopted to verify the validity of the CFM method, and the dynamical temporal behaviors of the transient response Probability Density Function (PDF) are analyzed. In addition, the transient bifurcation phenomenon of the system is discussed, and the 3-D distribution diagram of peak value of transient PDF is produced.

Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

This paper investigates the stochastic P-bifurcation (SPB) of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise. Variable transformation and the stochastic averaging technique are applied to derive the expression of probability density function (PDF) of the system response. Critical conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density function. Numerical results are given to show the effectiveness of the proposed approach. Stochastic P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions. Stochastic P-bifurcation (SPB) for additive noise can be investigated by the critical conditions. SPB induced by the fractional order is described in the bifurcation plane, see Fig. a. Figure b shows that the probability density function (PDF) for (0.1, 0.005) ∈ R2 has one peak, while the PDF for (0.8, 0.005) ∈ R2 has two peaks. It implies that the change of the fractional order leads to the variation of the structure of the PDF, which indicates the occurrence of the SPB.

Probabilistic response of a fractional-order hybrid vibration energy harvester driven by random excitation.

The stochastic response of a fractional-order hybrid vibration energy harvester is investigated in this paper. Equivalent system can be derived by the variable transformation. Then, the probability density functions of mechanical states are obtained by the stochastic averaging technique. The good agreement between numerical simulation and analytical results demonstrates the effectiveness of the proposed method. Mean square voltage, mean square current, and mean output power are presented to illustrate the device output performance. Results imply that the hybrid vibration energy harvesting system can generate higher mean output power than that from a separate piezoelectric system and an electromagnetic system.

Bifurcation Analysis of a Self-Sustained Birhythmic Oscillator Under Two Delays and Colored Noises

In this manuscript, an investigation on bifurcations induced by two delays and additive and multiplicative colored noises in a self-sustained birhythmic oscillator is presented, both theoretically and numerically, which serves for the purpose of unveiling extremely complicated nonlinear dynamics in various spheres, especially in biology. By utilizing the multiple scale expansion approach and stochastic averaging technique, the stationary probability density function (SPDF) of the amplitude is obtained for discussing stochastic bifurcations. With time delays, intensities and correlation time of noises regarded as bifurcation parameters, rich bifurcation arises. In the case of additive noise, it is identified that the bifurcations induced by the two delays are entirely distinct and longer velocity delay can accelerate the conversion rate of excited enzyme molecules. A novel type of P-bifurcation emerges from the process in the case of multiplicative colored noise, with the SPDF qualitatively changing between crater-like and bimodal distributions, while it cannot be generated when the multiplicative colored noise is coupled with additive noise. The feasibility and effectiveness of analytical methods are confirmed by the good consistency between theoretical and numerical solutions. This investigation may have practical applications in governing dynamical behaviors of birhythmic systems.

Influence of restricted operating space on the performance of random vibration energy harvesting

The ambition to create self-powered microscale electrical devices motivates scientific and industrial communities to investigate the energy harvesting technique, especially working in random vibration circumstances. The mechanical response of the random vibration system may approach infinity with small probability, and then the restricted operating space of the energy harvesting system will unavoidably induce the occurrence of collision interaction. Here, the random mechanical vibration and electrical output of the vibration energy harvesting system including inelastic collision are investigated, in which the random excitation is described by Gaussian white noise, while the collision interactions are described by the transient impact model and inelastic contact model, respectively. Introducing the generalized harmonic transformation of mechanical states and adopting a slow-varying process assumption of amplitude and averaged frequency, the output voltage can be explicitly expressed as the function of displacement, velocity, and system total energy by directly integrating the linear electrical equation. The transient impact interaction is equivalent to an effective damping with energy-dependent damping coefficient, while the inelastic contact interaction is equivalent to an effective damping and an affiliated potential energy. The averaged equations with respect to mechanical energy are then derived through the stochastic averaging technique. The stationary probabilistic density function of mechanical states is established by solving the reduced Fokker–Plank–Kolmogorov equation, and then the statistical quantities of electrical voltage are obtained by the relation between voltage and mechanical states. The effectiveness and precision of the analytical procedure are validated through the results from Monte Carlo simulations, and the influence of collision interaction on the performance of energy harvesting is discussed in detail. Also, for the energy harvesting system excited by colored noise, the influence of collision interaction on the performance is evaluated through Monte Carlo simulations.

Colored Noise in Oscillators. Phase-Amplitude Analysis and a Method to Avoid the itô-Stratonovich Dilemma

We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an Ornstein-Uhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a drift-diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of the previous phase reduced models.

Hilbert Transform–Based Stochastic Averaging Technique for Determining the Survival Probability of Nonlinear Oscillators

AbstractA Hilbert transform based definition of the response amplitude of randomly excited nonlinear oscillators has been proposed recently to address certain limitations associated with the standa...

Random vibration control for multi-degree-of-freedom mechanical systems with soft actuators

Soft robotics include soft actuators and possess the capacity of large deformation and environmental compatibility. Weak environmental disturbances may deteriorate the operating performance of soft robotics due to the low stiffness of soft actuators. This manuscript investigates multi-degree-of-freedom nonlinear mechanical systems with dielectric elastomer actuators and establishes the bounded/unbounded optimal control strategies to suppress the random vibration around the equilibrium position by adjusting the imposed voltage in real time. First, the constitutive relation of a plan-type dielectric elastomer actuator and then the vibrating equation of the multi-degree-of-freedom nonlinear system around the equilibrium position are derived successively. The bounded/unbounded optimal control problems are then established by adopting the corresponding performance indexes. The bounded/unbounded optimal control strategies are then derived by combining the stochastic averaging technique and stochastic dynamic programming principle. Numerical results on a two-degree-of-freedom nonlinear system illustrate the application and efficacy of the proposed optimal control strategies.

Stochastic averaging technique for SDOF strongly nonlinear systems with delayed feedback fractional-order PD controller

A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom (SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative (PD) controller. The delayed feedback fractional-order PD control force is approximated by an equivalent non-delay feedback control force combining with a quasi-linear elastic force and a quasi-linear damping force. The averaged Ito stochastic differential equation for amplitude of the equivalent nonlinear system is derived by the generalized harmonic functions. The analytical stationary probability density function (PDF) is obtained with solving the reduced Fokker-Planck-Kolmogorov (FPK) equation. Two examples of van der Pol oscillator and Rayleigh-Duffing oscillator are studied to illustrate the application and effectiveness of the proposed method. Numerical results display that the proposed method can yield to the high precision, and the time delay could ruin the control effectiveness, but also even amplifies the response of the system more than that of uncontrolled system. Furthermore, the study finds that the parameters of fractional-order α and time delay may cause the stochastic P-bifurcation. It is indicated that the delayed feedback fractional-order PD controller can offer a potentially effective tool for anti-control of stochastic bifurcation

Stochastic response of a vibro-impact system with variable mass

Vibro-impact machines are found increasingly broad application in various sectors of the world. And the mass disturbance exists in the process of impact, which refers to the variable mass system that is a common phenomenon in life. With the development of micro-electronic and nanotechnology, it is essential to investigate the effect of mass disturbance on this random vibro-impact system response. The stationary system response is the main topic of the manuscript. In the light of the discontinuity of vibro-impact system, the nonsmooth transformation is introduced to overcome the discontinuity. Then the stochastic averaging technique is applied to the transformed stochastic variable mass system. The approximate analytical response can be obtained by solving associated Fokker–Planck–Kolmogorov equation. A Van der Pol variable mass vibro-impact system is given to illustrate the application and availability of this method. Meanwhile, stochastic P-bifurcation is studied, and a reasonable interpretation is given to explain the procedure of stochastic P-bifurcation.

Stochastic averaging for bistable vibration energy harvesting system

Vibration energy harvesting technique provides the possibility of the development of self-sustaining microelectronic components. The bistable energy harvesting devices have higher efficiency than the traditional mono-stable devices and deserve further investigation. In this manuscript, a novel stochastic averaging procedure is established to evaluate the stationary random response of bistable energy harvester to additive and multiplicative white noises. An equivalent nonlinear system associated with the original coupling system is first derived by integrating the circuit equation and adopting the assumption of generalized harmonic functions of sample responses. The stationary probability density for mechanical energy of the equivalent nonlinear system is then derived through the stochastic averaging technique. The mean-square mechanical responses, the mean-square voltage and the mean output power are obtained analytically. Finally, the influences of crucial parameters, such as excitation intensity, coupling factor, time constant ratio etc., on mean-square voltage and mean output power are discussed in detail which may provide some guidance for structural design for maximizing output power.

A comparative study of the stochastic averaging method and the path integration method for nonlinear ship roll motion in random beam seas

In this paper, the energy-based stochastic averaging method and the path integration method are applied to study the stochastic response of the nonlinear roll motion in random beam seas. Specifically, the Markov diffusion theory is applied to describe the random roll motion such that the probabilistic properties of the ship roll motion are governed by the Fokker–Planck (FP) equation. The stochastic averaging method focuses on the roll energy envelope process and reduces the difficulty in calculating the stochastic response via a dimensional reduction of the original system. In contrast, the path integration method is based on the Markov property of the original dynamical system and provides approximate solutions to the FP equation by linking the explicitly known local solutions. Although the stochastic averaging technique is well established in the study of the stochastic responses of various dynamical systems, its accuracy for determining the high-level roll response has not yet been a focus of study. This paper aims to provide a comparative study for the performance of the two above-mentioned methods; additionally, the advantages and shortcomings of using each method for studying the nonlinear roll motion in beam seas are demonstrated by practical calculations.